ORBITAL OPTIMIZATION OF INTERPLANETARY TRAJECTORIES WITH ENVIRONMENTAL PERTURBATIONS AThesis presented to the Faculty of California Polytechnic State University, San Luis Obispo In Partial Fulfillment of the Requirements for the Degree Master of Science in Aerospace Engineering by Eric Woods June 2018 c 2018 Eric Woods ALL RIGHTS RESERVED ii COMMITTEE MEMBERSHIP TITLE: Orbital Optimization of Interplanetary Trajectories with Environmental Perturba- tions AUTHOR: Eric Woods DATE SUBMITTED: June 2018 COMMITTEE CHAIR: Kira Abercromby, Ph.D. Associate Professor of Aerospace Engineering COMMITTEE MEMBER: Colleen Kirk, Ph.D. Professor of Mathematics COMMITTEE MEMBER: Jordi Puig Suari, Ph.D. Professor of Aerospace Engineering COMMITTEE MEMBER: Eric Mehiel, Ph.D. Professor of Aerospace Engineering iii ABSTRACT Orbital Optimization of Interplanetary Trajectories with Environmental Perturbations Eric Woods For a detailed analysis of orbital optimization, it is desired to incorporate a space- craft environment model in order to have maximum confidence that the analysis will produce an accurate trajectory. Such a model requires the addition of orbital pertur- bations, or small forces acting on the spacecraft throughout its trajectory that can eventually accumulate in large distances over time. The optimization method that this thesis is concerned with is STOpS (Spacecraft Trajectory Optimization Suite), a Matlab optimizer created by Timothy J. Fitzgerald that utilizes an Island Model Paradigm with five di↵erent optimization algorithms. STOpS was originally built to model trajectories with the two body equations of motion. A Lambert’s method was utilized to link the spacecraft trajectory from planet to planet, and a flyby section was created for the hyperbolic gravity assist trajectories. A cost function was then used to evaluate the best combination of ∆V, time of flight, synodcity, flyby altitude, and heliocentric energy. This work is primarily concerned with adding the dynamics created by perturbations into Lambert’s problem as well as the gravity assist trajecto- ries. The improved analysis creates a more robust solution for dealing with optimized interplanetary trajectories. Two proven trajectories will be focused on for the main analysis of this thesis which are the trajectories taken by Voyager 2 in the tour of the solar system as well as Cassini’s mission to Saturn. When perturbations were added to the analysis of these missions, STOpS was able to find trajectories which met both ∆V and time of flight requirements for each mission. For the optimization of each of these missions the key dates of departure, flyby, and arrival at all the planets varied by no more than one year from the true trajectories of Voyager 2 and Cassini. iv ACKNOWLEDGMENTS Thanks to: IwouldfirstliketothankDr.Aforhercontinuedsupportthroughoutmy • graduate studies as my thesis advisor. I have learned a great deal from the many insights that she has provided over the years as a professor at Cal Poly. I would also like to thank all of the faculty in aerospace engineering as I would • not be where I am today without their continued knowledge and support. Finally, thank you to my family for supporting me throughout these long college • years and having the trust in me that I could accomplish all my goals as an aspiring engineer. vi TABLE OF CONTENTS Page LISTOFTABLES ................................. x LISTOFFIGURES ................................ xi CHAPTER 1Introduction................................... 1 1.1 Nomenclature ............................... 1 1.2 Statement of Problem . 2 1.3 Purpose of Study . 3 1.4 Existing Optimization Programs . 4 2ReviewofOrbitalMechanics.......................... 6 2.1 Elliptical Sections . 6 2.2 HyperbolicSections............................ 7 3Perturbations.................................. 8 3.1 Perturbations in the Spacecraft Environment . 8 3.1.1 N-bodyGravitation. 8 3.1.2 Solar Radiation Pressure . 9 3.1.3 Atmospheric Drag . 10 3.1.4 J2 ................................. 11 3.1.5 General Relativity . 12 3.2 Implementation . 13 4Optimization.................................. 14 4.1 Methodology . 14 4.2 Island Topology . 15 4.3 GeneticAlgorithms ............................ 17 4.3.1 Population Size . 18 4.3.2 Chance of Mutation . 18 4.3.3 NumberofGenerations. 19 4.4 Di↵erentialEvolution........................... 19 4.5 Particle Swarm Optimization . 20 vii 4.5.1 Particle Motion . 20 4.5.2 Informants............................. 21 4.6 AntColonyOptimization. 21 5Lambert’sProblem............................... 23 5.1 The Two Body Lambert’s Problem . 23 5.2 The Perturbed Lambert’s Problem . 26 5.2.1 Shepperd’sMethod . 26 5.2.2 Runge Kutta Fehlburg 78 . 32 5.2.3 Determination of the True Trajectory . 37 5.2.4 Consideration for Lambert’s Perturbations . 38 6 GravitationalAssists .............................. 40 6.1 TheTwoBodyGravityAssist . 40 6.2 ThePerturbedGravityAssist . 42 7 ResultsandAnalysis .............................. 47 7.1 Perturbation Magnitude Comparison . 47 7.2 AnalysisofTestCaseTrajectory. 48 7.3 Cassini Trajectory . 50 7.3.1 Cassini Background . 50 7.3.2 Cassini Optimization Setup . 51 7.3.3 CassiniAnalysisWithoutPerturbations . 54 7.3.4 CassiniAnalysisWithPerturbations . 59 7.4 Voyager 2 Trajectory . 62 7.4.1 Voyager 2 Background . 62 7.4.2 Voyager 2 Optimization Setup . 63 7.4.3 Voyager 2 Analysis Without Perturbations . 65 7.4.4 Voyager2AnalysisWithPerturbations . 68 8Conclusions................................... 72 8.1 Future Work . 74 BIBLIOGRAPHY ................................. 76 APPENDICES A STOpSUserGuide ............................ 78 A.1 OptimizationPanels ........................... 78 viii A.1.1 IslandTopology.......................... 78 A.1.2 TrajectoryInformation . 79 A.1.3 OptimizationOptions . 79 A.1.4 CostFunctionOptions . 79 A.1.5 ActualMission .......................... 80 A.2 ResultsofAnalysis ............................ 80 ix LIST OF TABLES Table Page 7.1 Perturbation Magnitude Comparison at One Astronomical Unit . 48 7.2 Flight Parameters of Trajectory From Earth to Saturn . 49 7.3 Cost Parameter Comparison for Cassini’s Unperturbed Trajectory . 57 7.4 Flight Time Comparison for Cassini’s Unperturbed Trajectory . 58 7.5 Cost Parameter Comparison for Cassini’s Perturbed Trajectory . 61 7.6 Flight Time Comparison for Cassini’s Perturbed Trajectory . 61 7.7 Cost Parameters for Voyager 2’s Unperturbed Trajectory . 67 7.8 Flight Times for Voyager 2’s Unperturbed Trajectory . 68 7.9 Cost Parameter Comparison for Voyager 2’s Perturbed Trajectory . 70 7.10 Flight Time Comparison for Voyager 2’s Perturbed Trajectory . 70 x LIST OF FIGURES Figure Page 4.1 Island Topology . 16 6.1 HyperbolicGravityAssistManeuver . 40 7.1 Trajectory From Earth to Saturn . 49 7.2 Cassini’s Trajectory Panel . 52 7.3 Cassini’s Cost Function Panel . 53 7.4 Cassini’s Optimization Panel . 54 7.5 Cassini’s Optimized Trajectory . 56 7.6 ActualCassiniTrajectory . 56 7.7 Cassini’s Optimized Trajectory With Perturbations . 60 7.8 Voyager 2 Trajectory Panel . 63 7.9 Voyager 2 Cost Function Panel . 64 7.10 Voyager 2’s Optimization Panel . 65 7.11 Voyager 2’s Optimized Trajectory . 66 7.12 Actual Voyager 2 Trajectory . 66 7.13 Voyager 2’s Optimized Trajectory With Perturbations . 69 xi Chapter 1 INTRODUCTION 1.1 Nomenclature ∆V change in velocity magnitude by spacecraft Ggravitationalconstant ~r position vector ~v velocity vector µ standard gravitational parameter Tperiodoforbit asemi-majoraxisoforbit Wsolarconstant cspeedoflight ⇢ atmospheric density CD coefficient of drag J2 zonal harmonic constant ✏ orbital energy ⇥statetransitionmatrix RSOI sphere of influence δ hyperbolic turn angle rp radius of perigee eeccentricity 1 1.2 Statement of Problem With any orbital trajectory there are an infinite number of paths that a spacecraft can take to get from point A to point B. While these trajectories may all be mathemat- ically feasible, they are not all equally useful for application in spacecraft missions. If it is decided that time of flight is an important parameter in a mission, then the trajectory that gets one spacecraft from Earth to Mars in 100 days is likely better than the one that takes 1 year to complete. This is where the method of optimization is particularly useful. Given any cost function, the user can decide which parameters are important to optimize and STOpS (Spacecraft Trajectory Optimization Suite) will search for a global minimum for this specified cost function. For this thesis, a test case trajectory will be studied from Earth to Saturn with a Jupiter gravity assist. The real spacecraft missions of Cassini as well as the trajectory taken by Voyager 2 on its tour of the solar system starting in August 20, 1977 will be also be analyzed in order to determine the accuracy of the optimization process developed in this work. In the test case trajectory, it is desired to find a feasible orbit that has been optimized to preserve as much fuel as possible while maintaining a relatively short time of flight. In the case of both Cassini and Voyager 2’s mission, these trajectories have already been well defined and proven to be successful. If this optimization process is to be shown to be robust, then STOpS should be able to calculate a trajectory that arrives at the same planets that both Cassini and Voyager 2 did with a similar ∆V and time of flight when compared to the actual spacecraft missions. The addition of pertur- bations should also produce a reasonable trajectory that agrees with the trajectory taken by Cassini and